Large time existence for 3D water-waves and asymptotics

By Borys Álvarez-Samaniego, David Lannes. 2007

323 citations recorded by the source corpus.

We rigorously justify in 3D the main asymptotic models used in coastal oceanography, including: shallow-water equations, Boussinesq systems, Kadomtsev–Petviashvili (KP) approximation, Green–Naghdi equations, Serre approximation and full-dispersion model. We first introduce a “variable” nondimensionalized version of the water-waves equations which vary from shallow to deep water, and which involves four dimensionless parameters. Using a nonlocal energy adapted to the equations, we can prove a well-posedness theorem, uniformly with respect to all the parameters. Its validity ranges therefore from shallow to deep-water, from small to large surface and bottom variations, and from fully to weakly transverse waves.The physical regimes corresponding to the aforementioned models can therefore be studied as particular cases; it turns out that the existence time and the energy bounds given by the theorem are always those needed to justify the asymptotic models. We can therefore derive and justify them in a systematic way.

Subfields: Applied Mathematics, Mathematical Physics, Oceanography

View Paper: https://www.semanticscholar.org/paper/13d1300f9ea4d1b219279db1caac769053f2d603

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In Applied Mathematics

In Mathematical Physics

In Oceanography

Citations in the corpus

A three-component generalization of Camassa–Holm equation with N-peakon solutions (2011, background): “In 1976, Green and Naghdi obtained a system of water wave equations to describe fluid flows in shallow water [27,1].”
Finite time singularities for the free boundary incompressible Euler equations (2011, background): “rface collapses at one point. 3 (a) t = T 1 (b) t = T 2 > T 1 Figure 2: Splat singularity. The interface collapses along an arc. The issue of long time existence has been treated in Alvarez-Lannes [3] where wellposedness over large time scales is shown and dierent asymptotic regimes are justied. ”
The Cauchy Problem for the Euler–Poisson System and Derivation of the Zakharov–Kuznetsov Equation (2012, result): “Contrary to the KP case (see [1] for the justification of various asymptotic models of surface waves) the rigorous justification of the long-wave limit of the EulerPoisson system has not been carried out so far (see however [5, 11] in the onedimensional case).”
Two dimensional water waves in holomorphic coordinates II: global solutions (2014, unknown): “Without being exhaustive, we list some of the more recent references [2, 1, 5, 6, 7, 12, 4, 13, 15, 16, 18, 20, 22].”
Asymptotic Models for Internal Waves (2007, methodology): “Together with consistency, a straightforward analysis would then provide a convergence result to the full Euler system, assuming that the large time existence results obtained by Alvarez-Samaniego and Lannes in [2] for the surface wave system are valid for the internal waves system.”
Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation (2009, methodology): “Th. 3.20 of [39] and Th. 3.1 of [ 29 ] where a similar formula is established for non necessarily flat surfaces).”
A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations (2007, methodology): “A rigorous derivation of the Green-Naghdi and Serre models is performed in [2], to which we refer for more details.”
Inverse scattering transform for the Degasperis–Procesi equation (2010, methodology): “In this parameter range, due to a balance between nonlinearity and dispersion, various integrable systems like the Korteweg–de Vries (KdV) equation arise as approximations to the governing equations [2].”
The Cauchy problem on large time for surface waves Boussinesq systems (2012, background): “This difficult step has been achieved in [1] in absence of surface tension (see [24] for gravity-capillary waves in the weakly transverse regime).”
Well‐posedness of Compressible Euler Equations in a Physical Vacuum (2010, background): “For multidimensional flows, there has been a lot of activity and important progress in free surface boundary problems, in particular regarding incompressible fluids [2–4, 11, 20, 37, 60, 64, 67].”
A new approach to handle wave breaking in fully non-linear Boussinesq models (2012, methodology): “…is, the error between the solution of the Green–Naghdi equations and the solution of the full Euler equations is then of order O(μ2t), as shown in Alvarez-Samaniego and Lannes (2008). dispersion properties, but also include a special treatment for wave breaking (classical Prandlt mixing length…”
Global solutions and asymptotic behavior for two dimensional gravity water waves (2013, result): “Moreover, we prove as well tame estimates, that complement similar results due to Craig, Schanz and Sulem (see [26], [60, Chapter 11] and [8, 43]), and establish bounds for the approximation of G(η)ψ (resp.”